Information Spreading Considering Repeated Judgment with Non-Recursion

Abstract

This paper investigates an information spreading mechanism under repeated judgment. In a generalized model, we prove that given a necessary condition, information under repeated judgment can sustain continuous spreading. Furthermore, we generalize the aforementioned spreading model on heterogeneous networks and calculate the analytic solution of the final state, in which spreaders finally have a stable scale to ensure that information can continuously spread when repeated judgment of information takes place. Moreover, the simulation results show that the more neighbors the spreaders have, the quicker the information vanishes. This finding suggests that in terms of information spreading under repeated judgement, it is not better to have more neighbors, quite contrary to common opinion.

1. Introduction

It is well-known that Nokia was, in the past, one of the best mobile brands. However, Nokia gradually lost its leading role to Apple and Samsung, who took full advantages of the smartphone trend. Currently, with the introduction of another trend of online direct sales, the competition in the mobile smartphone industry is increasingly more aggressive. To address this challenge, many suppliers face a channel distribution decision of whether to add a new retailing channel to their existing offline channels. For example, many top manufacturers, such as Apple, Cisco, Huawei, IBM, and Lenovo, have started selling directly online. What are the rationales behind such decisions of transition? In this paper, we adopt a perspective of how information spreading can help us to better understand business performance when making online/offline decisions. In this era of digital business, a crucial decision mainly depends on how much information you have, such as information about a corporation’s own strength, its rival’s reactions, and the market situation.

Digital business/economy has influenced human life greatly and is challenging traditional management theories. B2C websites allow consumers to comment on the products they have bought. Online product reviewing is one of the major components of the influence of word of mouth (IWOM), and it has a significant impact on the consumption decisions of online shoppers. Rational consumers want to maximize their utility, i.e., they consider relevant online product reviews before making a purchasing decision.

In this era of digital business, information is vital to any decision-making, and it is not an unfamiliar concept in ordinary daily life. For instance, smartphone, channel strategy, and WOM information is highly important in making decisions for both manufacturers and consumers. Obviously, information plays an important role in our daily life.

Information is a type of social contagion process through which the same or a similar remark spreads on a large scale in a short time through chains of interaction or communication. Although these problems are worthy of consideration, in this paper, a perspective of information spreading enabling us to focus on population dynamics is under investigation. Information preserves an asymptotic behavior that heavily depends on the scale of final spreaders. It is challenging to understand how information continuously spreads (see Figure 1 below).

We know the following with regard to information:

• Different cognitive degrees of information mainly depend on an individual’s educational level.
• Although it plays a critical role in purchasing decision, information value may not be fully recognized in terms of individual judgment.
• In initial judgment, users of information may find it useless, but the situation may change under repeated spreading (i.e., repeated judgment).

In the initial spreading stage of information, spreaders may behave with intense oscillation and frequent replacement. However, based on recursion, information may spread by a periodic orbit that causes the same information to spread around and begin again in one orbit, which may cease to spread eventually. How to ensure the continuous spread of information, with measures to implement repeated judgment for information and avoid recursive information, is the objective of this paper. In this study, we find that spreaders finally form a stable quantity in systems and obtain steady results from the organizational atmosphere.

As an example of social contagion processes, information spreading has attracted the interest of researchers from diverse disciplines [1,2,3,4]. Pioneering contributions to information spreading modeling based on epidemiological models date back to Trpevski et al. [5], who used the idea of information spreading to study how two different product competitors are expected to attract new users, and they found that a product with lower preference can still obtain a non-zero fraction of the market. Liu et al. [6] considered the characteristics of WOM communication and the heterogeneity of online social networks as scale-free networks. Miritello et al. [7] investigated the temporal patterns of human communication and its influence on the spreading of information in social networks and explained the opposite effects on the reach of the information, i.e., whereas bursts of anger hinder propagation at large scales, conversations favor local rapid cascades. Xu et al. [8] proposed an S-SEIR model that analyzes the influence of information value and user behavior on information dissemination on social networking services, and found that information value can benefit information audits within a certain range. Although these studies have mainly focused on the applications of information, some scholars have begun to analyze information itself via a physical model or the complex network theory [9,10,11]. For example, Cao et al. [9] established a general model by considering the level of forgetfulness during knowledge transmission in complex networks and analyzed the stability of the equilibrium points and the transmission threshold. From this angle, they used mathematical analysis through compartment models to explain information spreading. However, forgetting is an asymptotic process and its level mainly depends on the number of spreaders and the organizational atmosphere [9], and it is not completely included in the aforementioned studies.

2. Spreading Mechanism

Information spreading is similar to knowledge or rumor transmission with complex dynamic behavior. In this paper, we consider a closed system and an inhomogeneous mixed population with N individuals who can be divided into four different states, corresponding to ignorant, spreader, hibernator, and stifler, respectively.

• Ignorant. The ignorant state is comprised of individuals who have not heard of the information and are susceptible to be informed of this information.
• Spreader. Spreaders are those who spread the information.
• Hibernator. Hibernators consist of individuals who are transformed from being spreaders due to the forget mechanism and can be turned back into spreaders again based on organizational structuring. Furthermore, some hibernators will give up the information according to some reasons, such as the conflict between hibernators and other states with different cognitive degrees of information. On the other hand, information may lose its incumbent value. What is more, the rest still stays in a hibernating state.
• Stifler. Stiflers are composed of individuals who have heard of the information but do not spread it anymore.

The spreading process goes forward through direct/indirect contacts between spreaders and the rest of the population. Thus, it is possible to construct a social network G = (V, E), where V and E denote vertices and edges, respectively. The information spreading and state transition process is shown in Figure 2.

Spreading can occur when an ignorant and a spreader interact, and the ignorant does not stay in the ignorant state but chooses whether to spread the information or not. It mainly depends on the ignorant person’s cognitive degree of information. In addition, in the ignorant–stifler interaction, a stifler, whether interested in the information or not, will make up his own mind to confirm if the information contains the truth. In this paper, the stiflers are very active in talking about the information [12]. When the stifler talks about the information as a precaution against its spread, he may succeed in changing the ignorant into a stifler. On the other hand, the ignorant may doubt the stifler’s words and learn to believe the information. Moreover, spreaders spontaneously forget the information and switch to the hibernator state (temporary forgetfulness) resulting from the forget mechanism. When a hibernator interacts with a spreader, despite incumbent spreaders around, the hibernator may remind himself to recall the information. When a spreader contacts other spreaders or stiflers, the former may switch to a stifler state due to the conflict among different states with different cognitive degrees of information. The same may occur for a hibernator interacting with other hibernators or stiflers.

In summary, the interactions of information spreading in this model are defined as in

Table 1. Definition of the interactions during information spreading for state transition.

I+S	$\to$ S+S (I accepts the value of information)
	$\to$ R+S (I doubts the value of information)
I+R	$\to$ S+R (I doubts R’s words and accepts)
	$\to$ R+R (I is assimilated by R’s words)
S+S	$\to$ R+S (S disagrees with other S’s words due to contradictory information $\iff$ repeated judgement with information) $\to$ S+S (S agrees with other S’s words with reinforced information $\iff$ repeated judgement with information)
S+R	$\to$ R+R (S doubts the value of information under repeated judgement)
	$\to$ S+ S (R is assimilated by S’s words)
H+H	$\to$ R+H (H disagrees with other H’s words due to contradictory information $\iff$ repeated judgement with information) $\nrightarrow$ S+H (H agrees with other H’s words with recalled information, but H’s words do not have enough persuasiveness)
H+R	→ R+R (H believes R’s words under repeated judgement)
R+R	$\nrightarrow$ I+R (To avoid recursion, which stops information from continuously spreading, this case is not considered in the proposed model)
H+R	$\nrightarrow$ H+H (H does not have enough power to make R accept the value of the information, because the forget mechanism exists)

(I = ignorant, S = spreader, H = hibernator, R = stifler).

3. Methods—A Generalized Model

The contact rate is the frequency at which a spreader contacts others in a unit of time, commonly depending on the total number of populations $N$, denoted by $U\left(N\right)$. Spreading may occur if an ignorant interacts with a spreader. Assuming that the probability of contact spread is ${\beta }_0$, the contact rate is modified by β₀U(N), i.e., the adequate contact rate. The ratio of ignorants in the population is $\frac{I}{N}$ and a spreader’s average spreading rate to ignorants is $\frac{ {\beta }_{0}U\left(N\right)I}{N}$, then the number of new spreaders (i.e., the number of ignorants who are infected by all incumbent spreaders) at time $t$ is ${\beta }_{0}U\left(N\right)\frac{I\left(t\right)}{N\left(t\right)}S\left(t\right)$, which is referred to as incidence.

In fact, because the quantity at which a spreader contacts others in a unit of time is limited when the number of populations is very large ($N\to +\infty$), we can assume that the contact rate is a constant $k$ rather than $U\left(N\right)$. Furthermore, the effective contact rate can be redefined as $\beta ={\beta }_{0}k$, and information incidence is viewed as $\beta \frac{I\left(t\right)}{N\left(t\right)}S\left(t\right)$, which is defined as standard incidence. Our model adopts “standard incidence” throughout the paper, and

Table 2. Definitions for key parameters and dynamic variables.

Parameter	Definition
${\alpha }_1$:	the ignorant–spreader interaction parameter, which is the ratio of contact frequency or information spreading
${\alpha }_2$:	the ignorant–stifler interaction parameter, which is the ratio of contact frequency
${\theta }_1$:	the probability of an ignorant becoming a spreader after hearing the information, i.e., the ignorant believes the spreader’s words
${\theta }_2$:	the probability of an ignorant becoming a stifler after hearing the information, i.e., the ignorant disbelieves the spreader’s words
$\beta$:	the rate spreaders switch to stiflers through contact with other spreaders and stiflers
$\lambda$:	the rate hibernators switch to stiflers through contact with other hibernators and stiflers
$\mu$:	forgetful probability due to the forget mechanism
$\gamma$:	the probability of a temporary hibernator switching to being a spreader again based on the remember mechanism, i.e., organizational atmosphere

summarizes the resting parameters and decision variables in the proposed model.

3.1. Modeling and Analysis

Combining research background and spreading mechanism, the spreading model in the framework of differential dynamic system is as follows:

$$\{\begin{array}{ll}\frac{d}{dt}I(t)=\hfill & -{\alpha }_{1}I(t)\frac{S(t)}{N(t)}-{\alpha }_{2}I(t)\frac{R(t)}{N(t)}\hfill \\ \frac{d}{dt}S(t)=\hfill & {\alpha }_1{\theta }_{1}I(t)\frac{S(t)}{N(t)}-\beta S(t)\frac{S(t)+R(t)}{N(t)}-\mu S(t)\hfill \\ & +\gamma H(t)\frac{S(t)}{N(t)}+{\alpha }_2(1-{\theta }_2)I(t)\frac{R(t)}{N(t)}\hfill \\ \frac{d}{dt}H(t)=\hfill & \mu S(t)-\gamma H(t)\frac{S(t)}{N(t)}-\lambda H(t)\frac{H(t)+R(t)}{N(t)}\hfill \\ \frac{d}{dt}R(t)=\hfill & {\alpha }_1(1-{\theta }_1)I(t)\frac{S(t)}{N(t)}+\beta S(t)\frac{S(t)+R(t)}{N(t)}\hfill \\ & +\lambda H(t)\frac{H(t)+R(t)}{N(t)}+{\alpha }_2{\theta }_{2}I(t)\frac{R(t)}{N(t)}\hfill \end{array},$$

(1)

In order to carry out the mathematical analysis, we set the following assumptions:

(a)

${\alpha }_1$, ${\alpha }_2$ are strictly positive.

(b)

$0.5\le {\theta }_1\le 1$, $0\le {\theta }_2\le 0.5$, and ${\theta }_1+{\theta }_2>1$.

(c)

$\beta$, $\lambda$, $\mu$, $\gamma$ are nonnegative.

Following Driessche and Watmough [13], the progression from H to S and failures of transmission are not considered to be new spreaders, but rather the progression of a spreader through the various compartments. Hence,

$$ℱ=\left(\begin{array}{c}{\alpha }_1{\theta }_{1}IS+{\alpha }_2\left(1-{\theta }_2\right)IR\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\right)\mathrm{and}\mathcal{V}=\left(\begin{array}{c}\beta S\left(S+R\right)+\mu S-\gamma HS\\ \begin{array}{c}-\mu S+\gamma HS+\lambda H\left(H+R\right)\\ {\alpha }_{1}IS+{\alpha }_{2}IR\\ {\alpha }_1\left({\theta }_1-1\right)IS-\beta S\left(S+R\right)-\lambda H\left(H+R\right)-{\alpha }_2{\theta }_{2}IR\end{array}\end{array}\right),$$

where $ℱ$ is the rate of appearance of new spreaders, and $\mathcal{V}$ is the rate of transfer to other individuals.

The spreader compartments are H and S, giving n = 2. Without loss of generality, the point (S, H, I, R) = (0,0,1,0) is the information-free equilibrium (IFE). Then

$$F=\left(\begin{array}{cc}{\alpha }_1{\theta }_1& {\alpha }_2\left(1-{\theta }_2\right)\\ 0& 0\end{array}\right) \mathrm{and}V=\left(\begin{array}{cc}\mu & 0\\ {\alpha }_1\left({\theta }_1-1\right)& -{\alpha }_2{\theta }_2\end{array}\right),$$

giving

$$V^{-1}=\left(\begin{array}{cc}\frac{1}{\mu }& 0\\ \frac{{\alpha }_1\left({\theta }_1-1\right)}{\mu {\alpha }_2{\theta }_2}& -\frac{1}{{\alpha }_2{\theta }_2}\end{array}\right).$$

As a well-known concept, the basic reproduction number, denoted ${ℛ}_0$, is the ‘expected number of secondary cases produced, in a completely ignorant population, by a typical spreading individual’ [14]. Following Diekmann et al., we call the next generation matrix for the system (1) and set ${ℛ}_0=\rho \left(F{V}^{-1}\right)$, i.e., ${ℛ}_0$ is the maximum spectral radius of the matrix $F{V}^{-1}$. In addition, we can calculate the spread threshold of system (1) in the following way:

$${ℛ}_0=\underset{\lambda }{\max }\det \left(\lambda E-F{V}^{-1}\right)=\frac{{\alpha }_1\left({\theta }_1+{\theta }_2-1\right)}{\mu {\theta }_2}.$$

Proposition 1. 

If${ℛ}_0>1$, then the information-endemic equilibrium (IEE) of system (1) exists, which means that information has to be spread.

From Proposition 1, it could be verified when the information can achieve spreading. Furthermore, in order to depict information on how to perform repeated judgment in the spreading process, incumbent system needs to make some changes.

Let $x\left(t\right)$, $y_1\left(t\right)$, $y_2\left(t\right)$, and $z\left(t\right)$ be the density of Ignorant, Spreader, Hibernator, and Stifler, respectively, i.e.,

$$x\left(t\right)=\frac{I\left(t\right)}{N},$$

$$y_1\left(t\right)=\frac{S\left(t\right)}{N},$$

$$y_2\left(t\right)=\frac{H\left(t\right)}{N},$$

$$z\left(t\right)=\frac{R\left(t\right)}{N},$$

and $x\left(t\right)+y_1\left(t\right)+y_2\left(t\right)+z\left(t\right)=1$. Thus, the incumbent system (1) can be redefined as follows:

$$\{\begin{array}{ll}\frac{d}{dt}x(t)=& -{\alpha }_{1}x{y}_1-{\alpha }_{2}xz\\ \frac{d}{dt}{y}_1(t)= & {\alpha }_1{\theta }_{1}x{y}_1-\beta y_1(y_1+z)-\mu y_1\\ & +\gamma y_1{y}_2+{\alpha }_2(1-{\theta }_2)xz\\ \frac{d}{dt}{y}_2(t)= & \mu y_1-\gamma y_1{y}_2-\lambda y_2(y_2+z)\\ \frac{d}{dt}z(t)= & {\alpha }_1(1-{\theta }_1)x{y}_1+\beta y_1(y_1+z)\\ & +\lambda y_2(y_2+z)+{\alpha }_2{\theta }_{2}xz\end{array}.$$

(2)

Due to being a closed system, the above system is equivalent to a cubic system as follows:

$$\{\begin{array}{ll}\frac{d}{dt}{y}_1(t)& ={\alpha }_1{\theta }_1[1-(y_1+y_2+z)]y_1-\beta y_1(y_1+z)\\ & +{\alpha }_2(1-{\theta }_2)[1-(y_1+y_2+z)]z\\ & -\mu y_1+\gamma y_1{y}_2\\ \frac{d}{dt}{y}_2(t)& =\mu y_1-\gamma y_1{y}_2-\lambda y_2(y_2+z)\\ \frac{d}{dt}z(t)& ={\alpha }_1(1-{\theta }_1)[1-(y_1+y_2+z)]y_1\\ & +\beta y_1(y_1+z)+\lambda y_2(y_2+z)\\ & +{\alpha }_2{\theta }_2[1-(y_1+y_2+z)]z\end{array},$$

(3)

and a positive invariant set with respect to system (3) is

$$\Delta :=\left\{y_1, y_2, z\in {\mathbb{R}}^3|y_1\ge 0, y_2\ge 0, z\ge 0, y_1+y_2+z\le 1 \right\}$$

Furthermore, let ${\Delta }_0:=\left\{y_1,y_2,z\in {\mathbb{R}}^3|y_1>0,y_2>0,z>0,y_1+y_2+z\le 1\right\}$, and

$$\mathrm{f}\left(y_1,y_2,z\right)={\alpha }_1{\theta }_1\left[1-\left(y_1+y_2+z\right)\right]y_1-\beta y_1\left(y_1+z\right)-\mu y_1+\gamma y_1{y}_2$$

$$+{\alpha }_2\left(1-{\theta }_2\right)\left[1-\left(y_1+y_2+z\right)\right]z,$$

$$\mathrm{g}\left(y_1,y_2,z\right)=\mu y_1-\gamma y_1{y}_2-\lambda y_2\left(y_2+z\right),$$

$$\mathrm{h}\left(y_1,y_2,z\right)={\alpha }_1\left(1-{\theta }_1\right)\left[1-\left(y_1+y_2+z\right)\right]y_1+\beta y_1\left(y_1+z\right)+\lambda y_2\left(y_2+z\right)$$

$$+{\alpha }_2{\theta }_2\left[1-\left(y_1+y_2+z\right)\right]z,$$

$$\rho (y_1,y_2,z)=\frac{1}{y_1{y}_{2}z},$$

then we can calculate that

$$\frac{\partial }{\partial y_1}\left(\rho \mathrm{f}\right)+\frac{\partial }{\partial y_1}\left(\rho \mathrm{g}\right)+\frac{\partial }{\partial y_1}\left(\rho \mathrm{h}\right)$$

$$=-\frac{{\alpha }_1{\theta }_1+\beta }{y_{2}z}-\frac{{\alpha }_2\left(1-{\theta }_2\right)\left(1-y_2-z\right)}{y_1{}^2{y}_2}-\frac{\mu }{y_2{}^{2}z}-\frac{\lambda }{y_{1}z}$$

$$-\frac{{\alpha }_1\left(1-{\theta }_1\right)\left(1-y_1-y_2\right)}{y_2{z}^2}-\frac{\beta y_1}{y_2{z}^2}-\frac{\lambda y_2}{y_2{z}^2}<0.$$

Thus, the Dulac criterion [15] is applied to deduce that the domain ${\Delta }_0$ does not contain any periodic orbits or cyclic chains of system (3). Indeed, we can easily rule out the periodic orbits containing points on the coordinate axes. Then, the periodic orbits of system (3) are not contained in domain $\Delta$.

Proposition 2. 

Information can achieve sustained spread, i.e., information with repeated judgment really exists, which can result from system (3) that does not have any periodic orbits or cyclic chains in domain $\Delta$.

3.2. Threshold under Utility Function

Although we have used constant parameters in our models for simplicity, it is more realistic to define parameters as functions of population frequencies. In this section, we attempt to derive some complex relations between population frequencies with the forget mechanism and utility function that determine individual behavior.

A simpler form of utility function was first introduced in models with two products [16]. It has since been widely utilized in the economics, marketing, and operations management literature [17,18]. The utility function also encompasses the classical economic features of diminishing marginal rates of substitution and diminishing marginal utility. In this paper, the utility function of a representative consumer is expressed from the perspective of aggregated demand as follows (for more properties of this utility function, refer to the study by Ingene and Parry [19]):

$$U\equiv {\displaystyle \sum }_i\left({\delta }_i{D}_i-\frac{b{D}_i{}^2}{2}\right)-\tau {\displaystyle \sum }_{i\ne j}\frac{D_i{D}_j}{2}-{\displaystyle \sum }_i{p}_i{D}_i,$$

where $\tau \left(0\le \tau <1\right)$ denotes the product substitutability, ${\delta }_i$ is the nominal demand for product $i$ assuming the price is zero, $D_i$ is the realized demand of product $i$, and $p_i$ is the retail price of product $i$. This utility function also implicitly assumes that the consumer has sufficient budget to purchase the product.

As information can be viewed as a type of common product to be handled, the demand for information mainly depends on the quantity of different compartments (i.e., $I,S,H,\mathrm{and} R$). For this model-based approach, we simply assume the following:

(a)

utility decreasing of ignorants and utility increasing of stiflers are not considered due to double marginalization.

(b)

hibernators consider the forget mechanism.

Thus, we can modify the aforementioned utility function as follows:

$$U\equiv \left(\mu S-\frac{S^2}{2}\right)+\left(\gamma H-\frac{H^2}{2}\right)-\theta SH,$$

(4)

without loss of generality. Maximizing Equation (4) yields the utility performance for each compartment as follows:

$$U_S=\frac{\mu -\gamma \theta }{1-{\theta }^2},$$

$$U_H=\frac{\gamma -\mu \theta }{1-{\theta }^2}.$$

When an individual belongs to a large group, it would suffice for him, which motivates him to maintain and enlarge his group [12]. Considering a spreader decides whether to talk about the information with a partner whose state is unknown and assuming that the spreader will gain positive psychological expectation $b>0$ if the partner gives feedback, such as believing the information or being amused by it, the internal spreader would have enough power to attain other states to accept the information, or the external information is sufficiently “attractive” (believable, worthy spread, etc.). Conversely, the spreader gains negative psychological expectation $-c<0$, if the partner gives feedback, such as denying it or giving a different cognitive degree of information resulting in conflict. In fact, the spreader may deduce the partner’s feedback from the utility compartment $U_{\mathrm{S}}\mathrm{or}{U}_H$, and the attractiveness of the information is $p\in \left(0,1\right)$.

If the spreader successfully switches a susceptible listener to a new spreader with probability $p$, the utility of the spreader compartment $U_S$ will increase by 1, i.e., U_s + 1. However, when two spreaders interact, the latter may leave a spread compartment with probability $1-p$, and the utility of other compartments, except spread $U_H$, will increase by 1, i.e., $U_H+1$. In other cases, the utility $U_H$ will not change. The payoffs of one spreader are summarized in

Table 3. A payoff matrix for information spread.

Partner	Ignorant ($\mathit{I}$)	Spreaders ($\mathit{S}$)
Probability ($p$)	$b+\left(U_S+1\right)$	$U_S$
$\left(1-p\right)$	$-c+U_H$	$-c+\left(U_H+1\right)$
Partner	Hibernators ($H$)	Stiflers ($R$)
Probability ($p$)	$b+\left(U_S+1\right)$	$b+\left(U_S+1\right)$
(1 − ρ)	$-c+U_H$	−c + UH

.

If the spreader does not talk about the information, his psychological expectation will not change. If he talks about it, the expected increment of his psychological expectation considering the spreader’s utility is denoted by

$$\Delta =p\left\{\left[b+\left(U_S+1\right)\right]x+U_S{y}_1+\left[b+\left(U_S+1\right)\right]y_2+\left[b+\left(U_S+1\right)\right]\right\}$$

$$+\left(1-p\right)\left\{\left(-c+U_H\right)x+\left[-c+\left(U_H+1\right)\right]y_1+\left(-c+U_H\right)y_2+\left(-c+U_H\right)z\right\}$$

$$=p\left[b+\left(U_S+1\right)\right]+\left(1-p\right)\left[-c+U_H\right]+\left[1-\left(2+b\right)p\right]y_1.$$

Thus, we can obtain

$$I<\frac{p\left(1-{\theta }^2\right)\left(1+b+c\right)+p\left(1+\theta \right)\left(\mu -\gamma \right)+\left(\gamma -\mu \theta \right)-c\left(1-{\theta }^2\right)}{\left(1-{\theta }^2\right)\left[\left(2+b\right)p-1\right]}\times N$$

when $\Delta >0$.

Proposition 3. 

The supremum of spreaders ( $S$) is a necessary condition for repeated judgment of information. If all members of the population in the system have become spreaders, it is not necessary to spread, because all of them have heard about the information.

3.3. Numerical Simulations and Discussions

For information spreading, we have already analyzed its repeated judgment in the spreading process. Here, we will present numerical simulations to investigate the properties of information spreading model like system (1), and we assume that there are $N=1000$ (without loss of generality) individuals in this closed system. Moreover, the parameters are assumed as follows: ${\alpha }_1=0.05$ and ${\alpha }_2=0.2$ (the stiflers are very active to talk about the information as mentioned above); ${\theta }_1=0.9$ and ${\theta }_2=0.4$ according to the assumption; $\beta =\lambda =0.01$; $\mu =0.025$ due to the forget mechanism; and $\gamma =0.8$ as the result from the organizational atmosphere. At the same time, the initial values of system (1) are $\left(I,S,H,R\right)=\left(900,50,0,50\right)$ which are much closer to reality when ${ℛ}_0=1.5000$.

As depicted in Figure 3, due to the different interactions among different compartments, the spreaders will become stiflers at a given rate and all the ignorant persons will disappear, i.e., the entire population has heard the information. The major difference is whether to spread it or not. In the long run, the hibernators lastly have a stable scale, although the ratio of H is smaller. Moreover, this situation may guarantee an accelerated spread of information. Figure 3a depicts how spreaders change with repeated judgment. In the initial judgment, spreaders increase until a peak is reached. Conversely, the information comprehension of spreaders is more rational, which may result in frequent replacement until a stability is reached via a second repeated judgment. The more repeated the judgment of the information, the more stable the number of spreaders.

Figure 4a depicts that limit cycles (i.e., periodic orbits or cyclic chains) do not exist in the phase diagram of $S-H$, which is a necessary condition for the information’s repeated judgment. In addition, Figure 4b investigates how the final size of spreaders changes under different initial values. Obviously, the size of spreaders is asymptotically stable in the long term even though the initial quantities of spreaders are different. Figure 4b proves that the information-endemic equilibrium (IEE) is globally stable when ${ℛ}_0>1$.

Figure 5a depicts the influence of spread velocity on the spreaders’ scale, so we can adjust these two parameters to sustain the number of spreaders. Figure 5b reveals that the smaller $\mu$ is, the larger the number of spreaders is. After all, an organization has adequate power to maximize the recovery of the hibernator compartment. Figure 5b also depicts the influence of the forget mechanism on information spreading. Forget is an inevitable phenomenon; however, we can effectively control the forget probability $\mu$ to enhance information spreading. In short, the better the organizational atmosphere, the quicker the spreading velocity.

4. Methods—A Dynamic Model Considering Heterogeneous Networks

In Section 3, we prove the possibility of information’s repeated judgment. Giving the infimum ${ℛ}_0$ of system (1) via the next-generation matrix, we also adopt a utility function that considers a static game to obtain the supremum. Next, we generalize this model on heterogeneous networks to investigate the dynamic behaviors of information. Here, we theoretically calculate the analytical solution of the network model and provide some numerical simulations to further explain some management suggestions.

4.1. Abbreviations and Acronyms

By combining system (1) and complex networks, the mean-field equations can be redefined as follows:

$$\{\begin{array}{ll}\frac{d}{dt}{I}_k(t)& =-{\alpha }_1{I}_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{S_{k^{\prime }}(t)}{N_{k^{\prime }}}-{\alpha }_2{I}_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{R_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ \frac{d}{dt}{S}_k(t)& ={\alpha }_1{\theta }_1{I}_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{S_{k^{\prime }}(t)}{N_{k^{\prime }}}+\gamma H_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{S_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ & -\beta S_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{S_{k^{\prime }}(t)+R_{k^{\prime }}(t)}{N_{k^{\prime }}}-\mu S_k(t)\\ & +{\alpha }_2(1-{\theta }_2)I_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{R_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ \frac{d}{dt}{H}_k(t)& =\mu S_k(t)-\gamma H_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{S_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ & -\lambda H_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{H_{k^{\prime }}(t)+R_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ \frac{d}{dt}{R}_k(t)& ={\alpha }_1(1-{\theta }_1)I_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{S_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ & +\beta S_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{S_{k^{\prime }}(t)+R_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ & +\lambda H_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{H_{k^{\prime }}(t)+R_{k^{\prime }}(t)}{N_{k^{\prime }}}\\ & +{\alpha }_2{\theta }_2{I}_k(t)k{\displaystyle \sum _{k^{\prime }}}p\left(k^{\prime }\right|k)\frac{R_{k^{\prime }}(t)}{N_{k^{\prime }}}\end{array},$$

(5)

and conditional probability $p\left(k^{\prime }|k\right)$ is a given ignorant with $k$ which randomly chooses an edge and points to a spreader with $k^{\prime }$.

In a similar way, let $x_k$,$y_{1k}$, $y_{2k}$, and $z_k$ be the density of Ignorant, Spreader, Hibernator, and Stifler, respectively, i.e.,

$$x_k=\frac{I_k}{N_k},$$

$$y_{1k}=\frac{S_k}{N_k},$$

$$y_{2k}=\frac{H_k}{N_k},$$

$$z_k=\frac{R_k}{N_k},$$

and $x_k+y_{1k}+y_{2k}+z_k=1$, $I_k+S_k+H_k+R_k=N_k$, and ${{\displaystyle \sum }}^{ }{N}_k=N$. Thus, the incumbent system (5) can be redefined as follows:

$$\frac{d}{dt}{x}_k\left(t\right)=-{\alpha }_1{x}_k\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)y_{1k^{\prime }}\left(t\right)-{\alpha }_2{x}_k\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)z_{k^{\prime }}\left(t\right),$$

(6)

$$\begin{gathered} \frac{d}{dt} y_{1k}\left(t\right)={\alpha }_1{\theta }_1{x}_k\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)y_{1k^{\prime }}\left(t\right)-\beta y_{1k}\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)[y_{1k^{\prime }}\left(t\right)+z_{k^{\prime }}\left(t\right)]-\mu y_{1k}\left(t\right) \\ +\gamma y_{2k}\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)y_{1k^{\prime }}\left(t\right)+{\alpha }_2\left(1-{\theta }_2\right)x_k\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)z_{k^{\prime }}\left(t\right), \end{gathered}$$

(7)

$$\frac{d}{dt}{y}_{2k}\left(t\right)=\mu y_{1k}\left(t\right)-\gamma y_{2k}\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)y_{1k^{\prime }}\left(t\right)-\lambda y_{2k}\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)\left[y_{2k^{\prime }}\left(t\right)+z_{k^{\prime }}\left(t\right)\right],$$

(8)

$$\begin{gathered} \frac{d}{dt}{z}_k\left(t\right)={\alpha }_1\left(1-{\theta }_1\right)x_k\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)y_{1k^{\prime }}\left(t\right)+\beta y_{1k}\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)[y_{1k^{\prime }}\left(t\right)+z_{k^{\prime }}\left(t\right)] \\ +\lambda y_{2k}\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)[y_{2k^{\prime }}\left(t\right)+z_{k^{\prime }}\left(t\right)]+{\alpha }_2{\theta }_2{x}_k\left(t\right)k{\displaystyle \sum }_{k^{\prime }}p\left(k^{\prime }|k\right)z_{k^{\prime }}\left(t\right). \end{gathered}$$

(9)

Suppose the network is uncorrelated, then the conditional probability p(k′|k) is independent of k. Therefore, the conditional probability takes the form $p\left(k^{\prime }|k\right)=\frac{k^{\prime }p\left(k^{\prime }\right)}{k}$ [20], where $\langle k\rangle ={\displaystyle \sum }_{k}kp\left(k\right)$ is the mean-degree (or first moment) of the network.

For convenience of calculation, we introduce auxiliary functions as follows:

$$\{\begin{array}{l}q\left(k^{\prime }\right)=p\left(k^{\prime }\right|k)\\ {\varphi }_1(t)={\int }_0^t\sum y_{1k}\left(t^{\prime }\right)q(k)d{t}^{\prime }={\int }_0^t\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }\\ {\varphi }_2(t)={\int }_0^t\sum y_{2k}\left(t^{\prime }\right)q(k)d{t}^{\prime }={\int }_0^t\ll y_{2k}\left(t^{\prime }\right)\gg d{t}^{\prime }\\ \psi (t)={\int }_0^t\sum z_k\left(t^{\prime }\right)q(k)d{t}^{\prime }={\int }_0^t\ll z_k\left(t^{\prime }\right)\gg d{t}^{\prime }\end{array},$$

with the shortened form $\ll o_k\left(t\right)\gg ={\displaystyle \sum }_k{o}_k\left(t\right)q\left(k\right)$.

Equation (6) can be directly integrated as $x_k\left(t\right)=e^{-k\left[{\alpha }_1{\varphi }_1\left(t\right)+{\alpha }_2\psi \left(t\right)\right]}$, and let initial distribution $x_k\left(0\right)\approx 1$ without loss of generality. Next, multiply Equation (7) with $q\left(k\right)$, sum over $k$ and integrate the equation with respect to $t$, we can obtain

$$\frac{d{\varphi }_1\left(t\right)}{dt}={\alpha }_1{\theta }_1{{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }kq\left(k\right)x_k\ll y_{1k^{\prime }}\gg d{t}^{\prime }-\beta {{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }\ll k{y}_{1k}\gg \left[\ll y_{1k^{\prime }}\gg +\ll z_{k^{\prime }}\gg \right]d{t}^{\prime }-\mu {\varphi }_1\left(t\right)$$

$$+\gamma {{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }\ll k{y}_{2k}\gg \ll y_{1k^{\prime }}\gg d{t}^{\prime }+{\alpha }_2\left(1-{\theta }_2\right){{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }kq\left(k\right)x_k\ll z_{k^{\prime }}\gg d{t}^{\prime }$$

$$={\alpha }_1{\theta }_1{{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }kq\left(k\right)x_k\ll y_{1k^{\prime }}\gg d{t}^{\prime }-\beta {{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }\ll k{y}_{1k}\gg \left[\ll y_{1k^{\prime }}\gg +\ll z_{k^{\prime }}\gg \right]d{t}^{\prime }-\mu {\varphi }_1\left(t\right)$$

$$+\gamma {{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }\ll k{y}_{2k}\gg \ll y_{1k^{\prime }}\gg d{t}^{\prime }-{\alpha }_1\left(1-{\theta }_2\right){{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }kq\left(k\right)x_k\ll y_{1k^{\prime }}\gg d{t}^{\prime }+\left(1-{\theta }_2\right)\left[1-\ll x_k\gg \right].$$

As for Equation (9), in a similar way, we can also calculate

$$\frac{d\psi \left(t\right)}{dt}={\alpha }_1\left(1-{\theta }_1\right){{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }kq\left(k\right)x_k\ll y_{1k^{\prime }}\gg d{t}^{\prime }+\beta {{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }\ll k{y}_{1k}\gg \left[\ll y_{1k^{\prime }}\gg +\ll z_{k^{\prime }}\gg \right]d{t}^{\prime }$$

$$+\lambda {{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }\ll k{y}_{2k}\gg \left[\ll y_{2k^{\prime }}\gg +\ll z_{k^{\prime }}\gg \right]d{t}^{\prime }-{\alpha }_1{\theta }_2{{\displaystyle \int }}_0^t{{\displaystyle \sum }}^{ }kq\left(k\right)x_k\ll y_{1k^{\prime }}\gg d{t}^{\prime }+{\theta }_2\left[1-\ll x_k\gg \right].$$

In the long run, ${\varphi }_1\left(t\right)$ and $\psi \left(t\right)$ are constant. That means $\frac{d{\varphi }_1\left(t\right)}{dt}=0$ and $\frac{d\psi \left(t\right)}{dt}=0$ when $t\to +\infty$. For the above purpose, we can obtain

$$\begin{gathered} \gamma {{\displaystyle \int }}_0^{\infty }{{\displaystyle \sum }}^{ }\ll k{y}_{2k}\left(t^{\prime }\right)\gg \ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }-\mu {\varphi }_1(\infty )+[1-\ll x_k(\infty )\gg ] \\ +\lambda {{\displaystyle \int }}_0^{\infty }{{\displaystyle \sum }}^{ }\ll k{y}_{2k}\left(t^{\prime }\right)\gg \left[\ll y_{2k^{\prime }}\left(t^{\prime }\right)\gg +\ll z_{k^{\prime }}\left(t^{\prime }\right)\gg \right]d{t}^{\prime }=0, \end{gathered}$$

(10)

while assuming $\xi {{\displaystyle \int }}_0^{\infty }\ll k{y}_{2k}\left(t\right)\gg \left[1-\ll x_k\gg \right]dt-\mu {\varphi }_1(\infty )+\left[1-\ll x_k(\infty )\gg \right]=0$ for simplicity. From Equation (8)

$$\frac{d}{dt}{y}_{2k}\left(t\right)=\mu y_{1k}\left(t\right)-k\left\{\gamma \ll y_{1k^{\prime }}\gg +\lambda \left[\ll y_{2k^{\prime }}\gg +\ll z_{k^{\prime }}\gg \right]\right\}{y}_{2k}\left(t\right),$$

$$\frac{d}{dt}{y}_{2k}\left(t\right)+k\left\{\gamma \ll y_{1k^{\prime }}\gg +\lambda \left[\ll y_{2k^{\prime }}\gg +\ll z_{k^{\prime }}\gg \right]\right\}{y}_{2k}\left(t\right)=\mu y_{1k}\left(t\right),$$

$$\frac{d}{dt}{y}_{2k}\left(t\right)+\xi \left[1-\ll x_k\gg \right]y_{2k}\left(t\right)=\mu y_{1k}\left(t\right),$$

(11)

By solving the one order non-homogeneous differential equation, it directly leads to

$$y_{2k}\left(t\right)=\mu {{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)e^{\xi \left(t^{\prime }-t\right)+\xi \left\{{{\displaystyle \int }}_0^t\ll x_k\left(t^{\prime }\right)\gg d{t}^{\prime }-{{\displaystyle \int }}_0^{t^{\prime }}\ll x_k\left(t^{″}\right)\gg d{t}^{″}\right\}}d{t}^{\prime }=\mu {{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)e^{\xi \left(t^{\prime }-t\right)}dt\prime +o\left(\xi \right),$$

(12)

where $o\left(\xi \right)$ is the leading order of $y_{2k}\left(t\right)$ in $\xi$.

By using the theory of ordinary differential equations, Equations (7) and (8) can be directly solved in the following ways:

$$y_{1k}\left(t\right)={\alpha }_1{\theta }_{1}k{{\displaystyle \int }}_0^t{x}_k\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }-\beta k{{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)\left[\ll y_{1k}\left(t^{\prime }\right)\gg +\ll z_k\left(t^{\prime }\right)\gg \right]d{t}^{\prime }-\mu {{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)d{t}^{\prime }$$

$$+\gamma k{{\displaystyle \int }}_0^t{y}_{2k}\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }-\left(1-{\theta }_2\right){\alpha }_{1}k{{\displaystyle \int }}_0^t{x}_k\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }+\left(1-{\theta }_2\right)\left[1-x_k\left(t\right)\right]$$

$$={\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k{{\displaystyle \int }}_0^t{x}_k\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }-\mu {{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)d{t}^{\prime }+\gamma k{{\displaystyle \int }}_0^t{y}_{2k}\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }$$

$$+\left(1-{\theta }_2\right)\left[1-x_k\left(t\right)\right]+o\left(\beta \right),$$

$$y_{2k}\left(t\right)=\mu {{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)d{t}^{\prime }-\gamma k{{\displaystyle \int }}_0^t{y}_{2k}\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }-\lambda k{{\displaystyle \int }}_0^t{y}_{2k}\left(t^{\prime }\right)\left[\ll y_{2k}\left(t^{\prime }\right)\gg +\ll z_k\left(t^{\prime }\right)\gg \right]d{t}^{\prime }$$

$=\mu {{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)d{t}^{\prime }-\gamma k{{\displaystyle \int }}_0^t{y}_{2k}\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }+o\left(\lambda \right)$, furthermore,

$$y_{1k}\left(t\right)={\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k{{\displaystyle \int }}_0^t{x}_k\left(t^{\prime }\right)\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }-y_{2k}\left(t\right)+\left(1-{\theta }_2\right)\left[1-x_k\left(t\right)\right]+o\left(\beta \right)+o\left(\lambda \right)$$

(13)

$$={\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k{{\displaystyle \int }}_0^t\left[x_k\left(t^{\prime }\right)-1+1\right]\ll y_{1k}\left(t^{\prime }\right)\gg d{t}^{\prime }-y_{2k}\left(t\right)+\left(1-{\theta }_2\right)\left[1-x_k\left(t\right)\right]+o\left(\beta \right)+o\left(\lambda \right)$$

$$={\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k{{\displaystyle \int }}_0^t\left[x_k\left(t^{\prime }\right)-1\right]d{\varphi }_1\left(t^{\prime }\right)+{\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k{\varphi }_1\left(t\right)-y_{2k}\left(t\right)$$

$$+\left(1-{\theta }_2\right)\left[1-x_k\left(t\right)\right]+o\left(\beta \right)+o\left(\lambda \right)$$

$$={\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k\left\{\left[x_k\left(t\right)-1\right]{\varphi }_1\left(t\right)-{{\displaystyle \int }}_0^t{\varphi }_1\left(t^{\prime }\right)d\left[x_k\left(t^{\prime }\right)-1\right]\right\}+{\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k{\varphi }_1\left(t\right)-y_{2k}\left(t\right)$$

$$+\left(1-{\theta }_2\right)\left[1-x_k\left(t\right)\right]+o\left(\beta \right)+o\left(\lambda \right).$$

Inserting Equation (13) in Equation (12), we can derive the expression of $y_{2k}\left(t\right)$ as

$$\begin{gathered} y_{2k}\left(t\right)=\mu {{\displaystyle \int }}_0^t{y}_{1k}\left(t^{\prime }\right)e^{\xi \left(t^{\prime }-t\right)}d{t}^{\prime }+o\left(\xi \right) \\ =\mu {{\displaystyle \int }}_0^t\{{\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k\left\{\left[x_k\left(t^{\prime }\right)-1\right]{\varphi }_1\left(t^{\prime }\right)-{{\displaystyle \int }}_0^{t^{\prime }}{\varphi }_1\left(t^{″}\right)d\left[x_k\left(t^{″}\right)-1\right]\right\}+{\alpha }_1\left({\theta }_1+{\theta }_2-1\right)k{\varphi }_1\left(t^{\prime }\right) \end{gathered}$$

$$\begin{array}{ll}& -y_{2k}\left(t^{\prime }\right)+\left(1-{\theta }_2\right)\left[1-x_k\left(t^{\prime }\right)\right]\}{e}^{\xi \left(t^{\prime }-t\right)}d{t}^{\prime }+o\left(\xi \right)+o\left(\beta \right)+o\left(\lambda \right)\\ =-\mu {\alpha }_1({\theta }_1+{\theta }_2-1)& k^2[{\varphi }_1{-}_1(\infty )+{\alpha }_2\psi (\infty )]{\varphi }_1(\infty )\times {\int }_0^t\frac{1}{2}[f^2\left(t^{\prime }\right)+f^2(0)]e^{\xi \left(t^{\prime }\varphi t\right)}d{t}^{\prime }\\ & +\mu k\{{\alpha }_1({\theta }_1+{\theta }_2-1){\varphi }_1(\infty )\\ & +(1-{\theta }_2)[{\alpha }_1{\varphi }_1(\infty )+{\alpha }_2\psi (\infty )]\}{\int }_0^{t}f\left(t^{\prime }\right)e^{\xi (t^{\prime }-t)}d{t}^{\prime }+o(\xi )+o(\beta )+o(\lambda )\\ & +o(\mu ),\end{array}$$

(14)

and $1-x_k\left(t\right)=1-{\mathrm{e}}^{-k\left[{\alpha }_1{\varphi }_1\left(t\right)+{\alpha }_2\psi \left(t\right)\right]}=1-{\displaystyle \sum }_n\frac{{\left\{-k\left[{\alpha }_1{\varphi }_1\left(t\right)+{\alpha }_2\psi \left(t\right)\right]\right\}}^n}{n!}=1-\left\{1-k\left[{\alpha }_1{\varphi }_1\left(t\right)+{\alpha }_2\psi \left(t\right)+o\right]\right\}=k\left[{\alpha }_1{\varphi }_1\left(t\right)+{\alpha }_2\psi \left(t\right)\right]$ with Taylor expansion. ${\varphi }_1\left(t\right)$, $\psi \left(t\right)$ will be in stationary solution, on the other hand, when close to the spread threshold ${ℛ}_0$. Asserting ${\varphi }_1\left(t\right)={\varphi }_1(\infty )f\left(t\right)$ and $\psi \left(t\right)=\psi (\infty )f\left(t\right)$, where $f\left(t\right)$ is a finite function.

Afterwards, inserting Equation (14) in Equation (10) to solve ${\varphi }_1(\infty )$

$$\begin{gathered} \xi {{\displaystyle \int }}_0^{\infty }{{\displaystyle \sum }}^{ }\ll k{y}_{2k}\left(t\right)\ll [1-\ll x_k(\infty )\gg ]dt-\mu {\varphi }_1(\infty )+\left[1-\ll x_k(\infty )\gg \right]=0, \\ \xi \frac{{\langle k\rangle }^2}{\langle k\rangle }\left[{\alpha }_1{\varphi }_1(\infty )+{\alpha }_2\psi (\infty )\right]{{\displaystyle \int }}_0^{\infty }\ll k{y}_{2k}\left(t\right)\gg f\left(t\right)dt-\mu {\varphi }_1(\infty )+\frac{{(\infty )}^2}{(\infty )}\left[{\alpha }_1{\varphi }_1(\infty )+{\alpha }_2\psi (\infty )\right]=0, \end{gathered}$$

$$\begin{gathered} -\mu {\alpha }_1\left({\theta }_1+{\theta }_2-1\right)\xi {\langle k\rangle }^4\left[{\alpha }_1{\varphi }_1(\infty )+{\alpha }_2\psi (\infty )\right]{\varphi }_1(\infty )A \\ +\mu \xi {\langle k\rangle }^3\left\{{\alpha }_1\left({\theta }_1+{\theta }_2-1\right){\varphi }_1(\infty )+\left(1-{\theta }_2\right)\left[{\alpha }_1{\varphi }_1(\infty )+{\alpha }_2\psi (\infty )\right]\right\}B \\ =\frac{\mu {\varphi }_1(\infty ){\langle k\rangle }^2}{\left[{\alpha }_1{\varphi }_1(\infty )+{\alpha }_2\psi (\infty )\right]\langle k^2\rangle }-\langle k\rangle , \end{gathered}$$

(15)

and $1-\ll x_k\left(t\right)\gg =\left(\langle k^2\rangle /\langle k\rangle \right)\left[{\alpha }_1{\varphi }_1\left(t\right)+{\alpha }_2\psi \left(t\right)\right]$, $A={{\displaystyle \int }}_0^{\infty }f\left(t\right){{\displaystyle \int }}_0^t\frac{1}{2}\left[f^2\left(t^{\prime }\right)+f^2\left(0\right)\right]·$ $e^{\xi \left(t^{\prime }-t\right)}d{t}^{\prime }dt$, $B={{\displaystyle \int }}_0^{\infty }f\left(t\right){{\displaystyle \int }}_0^{t}f\left(t^{\prime }\right)e^{\xi \left(t^{\prime }-t\right)}d{t}^{\prime }dt$.

In an ideal case, we may assume that ${\varphi }_1(\infty )+\psi (\infty )=1$, i.e., all of the people have heard of the information, and the key lies in whether spreading it or not. Thus, we can calculate ${\varphi }_1(\infty )$ and $\psi (\infty )$, respectively;

4.2. Numerical Simulations and Discussions

This section will present some numerical simulations to interpret the dynamic behaviors of the system (5). Additionally, the simulations are based on a ‘scale-free’ network [21] with the power-law distribution takes the form $p\left(k\right)=\left({\gamma }_0-1\right)m^{\left({\gamma }_0-1\right)}{k}^{-{\gamma }_0}$, where ${\gamma }_0=3$, and $m$ is the minimum degree of the network. The parameters’ setting is same as Section 3.3.

Figure 6a depicts the influence of the neighbors of spreaders on information spreading. It is obvious that the more the neighbors of spreaders, the less the scale of the final spreaders. Contradictions increase when more people are around the spreaders, thus the quantity of spreaders leads to intense oscillation, and the system (5) is more vulnerable. Over time, the spreaders finally obtain a stable scale to ensure the continuous spread of information when a repeated judgment of information takes place, which is depicted in Figure 6b. Moreover, these numerical results can be verified by the analytical solution of Equation (15).

As administrators, it is more important to set up effective neighbors of information management, rather than to adopt the idea of the more the better.

5. Conclusions

This study investigates the impact of repeated judgment and the forgetting mechanism on information spreading. Unlike previous studies on information spreading, our study concentrates on the mathematical analysis of a network model and provides some necessary conditions to explain how information continuously spread.

(a)

For continuous inheritance and innovation of information, ${ℛ}_0$ as a spread threshold has to meet ${ℛ}_0>1$, which mainly depends on the definition of the basic reproduction number; otherwise, it will completely extinct.

(b)

As rational consumers, people always expect to maximize their utility whenever they purchase. Based on this theory, we apply the utility function to obtain spreaders’ supremum $S_{max}$.

(c)

Spreading can occur only when spreaders and other individuals contact and interact, implying that individuals from different orbits need to interact. Thus, periodic orbits or cyclic chains do not exist; otherwise, we cannot carry out repeated judgments of information.

When conditions (a) and (b) are satisfied, information undoubtedly can spread. Information value is manifested during the process of information spread. As for the information itself, it demonstrates different behaviors under different spreading stages, and Figure 1 depicts some interpretations in detail. In Figure 1, we take the repeated judgment of information as an example, and the reason why this phenomenon occurs is non-recursion of information judgment, i.e., no periodic orbits or cyclic chains exist in our model.

Forget is an inherent characteristic of information, which is the so-called vulnerability. It is worthwhile to enhance the robustness of the information-spreading process. From a technical perspective, we can strengthen repeated judgment of information. In addition, an organization as a whole may build a long-term effective learning atmosphere. One way is to encourage active efforts and self-improvement of members, and enhancing active communications among members is another way. In conclusion, we must establish an effective information judgment mechanism in the organization.

In the network model of information spreading, the number of neighbors of a node has a huge impact on information spreading, except for the forgetting mechanism, i.e., as the number of neighbors around one node increases, the spread velocity decreases. This is because conflicts are exposed when more neighbors are around one node, which results in more people choosing to give up the information. Then, as a rational administrator, one needs to effectively control the above phenomenon to implement information management. After all, the number of neighbors of one people is not the more the better, contrary to the common sense of the principle of “six degrees of separation”.

People expect information (distinct from rumors) to have a faster spreading velocity, so it can help an organization to establish a better learning atmosphere. It can also gain advantages in making decisions for better operational management.